5.16 Radius and interval of convergence with power series

Radius and interval of convergence with power series

Lessons

Notes:
Note *Power Series are in the form:

n=0cn(xa)n \sum_{n=0}^{\infty}c_n(x-a)^n

where cnc_n are the coefficients of each term in the series and aa is number.

To find the Radius of Convergence of a power series, we need to use the ratio test or the root test. Let An=cn(xa)nA_n=c_n(x-a)^n. Then recall that the ratio test is:

L=limL=\limn →\inftyAn+1An|\frac{A_{n+1}}{A_n}|
and the root test is
L=limL=\limn →\inftyAn1n|A_n|^{\frac{1}{n}}

where the convergence happens at LL< 11 for both tests. More accurately we can say that the convergence happens when xa|x-a| < RR, where is the Radius of Convergence.

The Interval of Convergence is the value of all xx’s, for which the power series converges. So it is important to also check if the power series converges as well at xa=R|x-a|=R.
  • 1.
    Radius and Interval of Convergence with Power Series Overview
  • 2.
    Questions based on Radius of Convergence
    Determine the radius of convergence for the following power series:
  • 3.
    Radius of Convergence of Sine and Cosine
    Determine the radius of convergence for the following power series:
  • 4.
    Questions based on Interval of Convergence
    Determine the interval of convergence for the foll owing power series:
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Radius and interval of convergence with power series

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